Theresa May has suffered a major embarrassment in the House of Commons after the Democratic Unionist party backed Labour motions in favour of increasing NHS pay and against a rise in tuition fees.

Labour’s motions passed on Wednesday without being pushed to a vote after it became clear the government had no majority to oppose the call for an end to the public sector pay cap for NHS workers nor the £250 a year increase in student fees.

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It is the first example of the DUP breaking with May since they struck a confidence and supply agreement to vote together on crucial legislation after the general election.

The motions fell outside the Tory-DUP deal as they were not binding, but their passage was nevertheless a symbolic victory for Labour and a sign that there is no longer a majority in the House of Commons for many of the austerity policies introduced by the Conservatives.

The DUP spoke out in favour of the pay motion and MPs also indicated the party backed a second motion tabled by Labour opposing a £250 a year rise in the cap on tuition fees to £9,250, which the Conservatives have passed by statutory instrument without a Commons debate despite protests from the opposition.

The DUP’s decision to side with Labour forced the Conservatives to abstain in an attempt to take the sting from the situation.

It will have no direct effect on policy, but is a sign that Labour may be able to exploit differences between the Conservatives and the DUP in the future in order to inflict more meaningful defeats on May.

Angela Rayner, the shadow education secretary, said the refusal to allow a binding vote on tuition fees had created a constitutional crisis. She said: “The government had no mandate to increase fees to begin with and if they do not now reverse the fee hike they will be defying the will of parliament in blatant disregard for our democracy”.

The DUP’s move is in line with its longstanding approach to wanting to lift the pay cap; it backed a similar early day motion earlier this year.

The party is also opposed to higher student fees, having voted against lifting the cap to £9,000 under the coalition.

Jeffrey Donaldson, the DUP MP for Lagan Valley MP, denied that supporting Labour on NHS wages and opposing tuition fees put the party’s deal that props up May’s government at peril.

“No, not at all,” Donaldson told the Guardian. “We made clear to her majesty’s government on issues like this we reserve the right to vote on the basis of our own manifesto. This doesn’t threaten the deal at all.”

Ian Paisley, the DUP MP for North Antrim, said he hoped the vote would be a “clarion call” from the Commons for fairer pay for NHS workers.

Downing Street has already signalled it will be “flexible” over the public sector pay cap in future, after coming under huge pressure over the issue since the election, when May lost her majority.

The prime minister initially held out against changing the policy but some Conservatives MPs in marginal seats lobbied for an end to seven years of austerity. May will need DUP votes in order for the budget to get the approval of parliament this autumn, so ending the cap may have been a political necessity.

Earlier this week, No 10 said the pay cap would end for prison officers, who will get a 1.7% pay rise, and police, who get a 2% pay increase next year, while signalling that the overall cap of 1% is likely to be eased at the next budget.

However, Labour and the unions have criticised the promised end to the pay cap as insufficient. In the Commons, Jonathan Ashworth, the shadow health secretary, said the rise amounted and amount to a pay cut with inflation is running at 2.9%.

“If the government are indeed abandoning this cap, let us put them on notice,” he said. “It must apply to the whole of the public sector, including the 55% of workers who are not covered by the pay review bodies.

“And we also put them on warning: we will not accept a divide-and-rule approach. We will not accept playing one set of public sector workers off against another.”

Ashworth said it “wasn’t good enough” for ministers to grant more flexibility over pay and expect hospitals to fund a wage increase for staff from existing budgets.

MPs told the House of Commons about examples of public sector workers, including ambulance technicians and nurses who are struggling to get by, being forced into debt or to take second jobs.

Following the debate, Ashworth called on May to abide by the wishes of the House of Commons and deliver higher pay for NHS workers. “The real question is, will the government now ignore the clear will of the house or will it take action to end the pay cap in the NHS?” he said.

Unison, the trade union for public sector workers, said the passage of the motion showed the Conservatives had lost the argument on public sector pay.

“Ministers must know they’re in the wrong when even their quasi-coalition partners in the DUP have turned against them, and backed pay rises for NHS workers,” said Dave Prentis, the general secretary of Unison.

“Theresa May is losing her majority on public sector pay, and we know there are Conservative MPs sympathetic to our calls for real pay rises for all public service workers.

“It’s now time for the prime minister and the chancellor to deliver those proper pay rises, and not drag this out a day longer. We’ll be keeping up the pressure on all MPs until the vote, and until all public servants get the pay rises they deserve.”